New conserved integrals and invariants of radial compressible flow in $n>1$ dimensions

TL;DR

This paper identifies new conserved integrals and invariants for radial compressible flow in multiple dimensions, including hierarchies of higher-order invariants and hidden conserved quantities, expanding understanding of flow invariants.

Contribution

It discovers three new invariants beyond entropy, introduces a recursion operator for generating higher-order invariants, and uncovers hidden conserved integrals in radial compressible flow.

Findings

Found three additional invariants besides entropy.

Developed a recursion operator for invariants hierarchies.

Uncovered hidden conserved integrals related to enthalpy-flux and entropy-weighted energy.

Abstract

Conserved integrals and invariants (advected scalars) are studied for the equations of radial compressible fluid/gas flow in $n>1$ dimensions. Apart from entropy, which is a well-know invariant, three additional invariants are found from an explicit determination of invariants up to first-order. One holds for a general equation of state, and the two others hold only for entropic equations of state. A recursion operator on invariants is presented, which produces two hierarchies of higher-order invariants. Each invariant yields a corresponding integral invariant, describing an advected conserved integral on transported radial domains. In addition, a direct determination of kinematic conserved densities uncovers two "hidden" non-advected conserved integrals: one describes enthalpy-flux, holding for barotropic equations of state; the other describes entropy-weighted energy, holding for entropic equations of state. A further explicit determination of a class of first-order conserved densities shows that the corresponding non-kinematic conserved integrals on transported radial domains are equivalent to integral invariants, modulo trivial densities.